1,721,076 research outputs found
On the equivalence between some projected and modulus-based splitting methods for linear complementarity problems
No full textIn this paper, we analyze the relationship between projected and (possibly accelerated) modulus-based matrix splitting methods for linear complementarity problems. In particular, first we show that some well-known projected splitting methods are equivalent, iteration by iteration, to some (accelerated) modulus-based matrix splitting methods with a specific choice of the parameter Ω. We then generalize this result to any Ω by formulating new classes of projected splitting methods and also provide a formal projection-based formulation for general (accelerated) modulus-based matrix splitting methods. Finally, we introduce and solve several test problems to evaluate also numerically the equivalence between the analyzed methods
A modulus-based framework for weighted horizontal linear complementarity problems
Full text linkWe develop a modulus-based framework to solve weighted horizontal linear complementarity problems (WHLCPs). First, we reformulate the WHLCP as a modulus-based system whose solution, in general, is not unique. We characterize the solutions by discussing their sign pattern and how they are linked to one another. After this analysis, we exploit the modulus-based formulation to develop new solution methods. In particular, we present a non-smooth Newton iteration and a matrix splitting method for solving WHLCPs. We prove the local convergence of both methods under some assumptions. Finally, we solve numerical experiments involving symmetric and non-symmetric matrices. In this context, we compare our approaches with a recently proposed smoothing Newton's method. The experiments include problems taken from the literature. We also provide numerical insights on relevant parts of the algorithms, such as convergence, attraction basin, and starting iterate
On the solution of general absolute value equations
No full textIn this note, we provide necessary and sufficient conditions that ensure the existence and uniqueness of solution of the general form of absolute value equations (AVEs), Ax−B|x|=b. The performed analysis is based on the equivalence between AVEs and horizontal linear complementarity problems (HLCPs). New sufficient conditions are proposed as well. We then compare the proposed conditions with recent results in the literature and we detail how efficient solution methods for HLCPs can be easily applied to the solution of general AVEs. Finally, we provide comments on the solvability of general AVEs under conditions larger than uniqueness of solution
A modulus-based formulation for the vertical linear complementarity problem
No full textWe introduce a modulus-based formulation for vertical linear complementarity problems (VLCPs) with an arbitrary number l of matrices. This formulation can be used to set up a variety of modulus-based solution methods, including, for example, the modulus-based matrix splitting methods for VLCPs that we here introduce. In this context, we particularly analyze the methods for problems with l = 2 (providing also sufficient conditions for their global convergence) and we then generalize the formulation of the methods to any l. Finally, some numerical experiments are solved to evaluate the performance of the proposed methods, which we compare with an existing smoothing Newton method for VLCPs
Modulus‐based synchronous multisplitting methods for solving horizontal linear complementarity problems on parallel computers
No full textIn this article, we generalize modulus-based synchronous multisplitting methods to horizontal linear complementarity problems. In particular, first we define the methods of our concern and prove their convergence under suitable smoothness assumptions. Particular attention is devoted also to modulus-based multisplitting accelerated overrelaxation methods. Then, as multisplitting methods are well-suited for parallel computations, we analyze the parallel behavior of the proposed procedures. In particular, we do so by solving various test problems by a parallel implementation of our multisplitting methods. In this context, we carry out parallel computations on GPU with CUDA
A nonlinearity lagging method for non-steady diffusion equations with nonlinear convection terms
Full text linkWe analyze an iterative procedure for solving nonlinear algebraic systems arising from the discretization of nonlinear, non-steady reaction-convection-diffusion equations with non-constant (and, in general, nonlinear) velocity terms. The basic idea underlying the procedure consists in lagging the diffusion and the velocity terms of the discretized system, which is thus partly linearized. After analyzing the discretized system and proving some results on the monotonicity of the operators and on the uniqueness of the solution, we prove sufficient conditions that ensure the convergence of this lagged method. We also describe the inner iteration and show how the weakly nonlinear systems arising at each lagged iteration can be solved efficiently. Finally, we analyze numerically the entire solution process by several numerical experiments
A modulus-based nonsmooth Newton’s method for solving horizontal linear complementarity problems
No full textModulus-based matrix splitting algorithms for generalized complex-valued horizontal linear complementarity problems
Full text linkIn this paper, we introduce the complex-valued horizontal linear complementarity problem (CHLCP), we provide two equivalent real-valued reformulations, and study modulus-based matrix splitting algorithms for solving the CHLCP. This latter point is motivated by the recent introduction of modulus-based matrix splitting methods for (non-horizontal) complex linear complementarity problems (CLCPs), which we generalize. We study the convergence of the proposed algorithms. Whenever possible, we seek convergence conditions that are directly based on the form of the real and imaginary parts of the matrices of the CHLCP in its complex form. This makes the convergence easier to evaluate than in existing convergence analyses. Finally, we study the numerical properties of the proposed algorithms by solving several CHLCPs. In this context, we also revisit results on the CLCP under the larger CHLCP framework, providing new numerical insights on the efficiency of existing algorithms for the CLCP
Temperature-induced neutral-to-ionic phase transition of the charge-transfer crystal tetrathiafulvalene-fluoranil
No full textThe temperature induced neutral to ionic phase transition (TI-NIT) is a rare phenomenon occurring in mixed stack charge transfer (CT) crystals made up of alternating π-electron donor (D) and acceptor (A) molecules. We were able to grow crystals of tetrathiafulvalene-fluoranil (TTF-FA), and to show that it undergoes TI-NIT like the prototype CT crystal TTF-chloranil. We characterized both room and low-T phases through IR and Raman spectroscopy and x-ray diffraction, demonstrating that while TTF-FA is quasineutral at room temperature, its ionicity jumps from 0.15 to 0.7 at low T, therefore crossing the neutral-ionic borderline. The transition, occurring around 150 K, is first order, with large thermal hysteresis and accompanied by crystal cracking. In the high-T phase D and A molecules lie on inversion center, i.e., the stacks are regular, whereas the low-T phase is characterized by the loss of the inversion symmetry along the stack as the stacks are strongly dimerized and by the doubling of the unit cell
A Framework for Physics-Informed Deep Learning Over Freeform Domains
No full textDeep learning is a popular approach for approximating the solutions to partial differential equations (PDEs) over different material parameters and boundary conditions. However, no work has yet been reported on learning PDE solutions over changing shapes of the underlying domain. We present a framework to train neural networks (NN) and physics-informed neural networks (PINNs) to learn the solutions to PDEs defined over varying freeform domains. This is made possible through our adoption of a parametric non-uniform rational B-Spline (NURBS) representation of the underlying physical shape. Distinct physical domains are mapped to a common parametric space via NURBS parameterization. In our approach, we formulate NNs and PINNs that learn the solutions to PDEs as a function of the shape of the domain itself through shape parameters. Under this formulation, the loss function is based on an unchanging parametric domain that maps to a variable physical domain. Residual computation in PINNs is made possible through the Jacobian of the mapping. Numerical results show that our networks can be trained to predict the solutions to a PDE defined over an entire set of shapes. We focus on the linear elasticity PDE and show how we can build a surrogate model that is able to predict displacements and stresses over a variety of freeform domains. To assess the efficacy of all networks in this work, data efficiency, network accuracy, and the capacity of networks to extrapolate are considered and compared between NNs and PINNs. The comparison includes cases where little training data is available. Transfer learning and applications to shape optimization are analyzed as well. A selection of the used codes and datasets is provided at https://github.com/fmezzadri/shape_parameterized.git
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